A note on twisted discrete singular Radon transforms

Abstract

In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an ${\ell }^p$ result for translation invariant discrete singular Radon transforms to a class of twisted operators including an additional oscillatory component, via a simple method of descent argument. Second, we note an ${\ell }^2$ bound for quasi-translation invariant discrete twisted Radon transforms. Finally, we extend an existing ${\ell }^2$ bound for a closely related non-translation invariant discrete oscillatory integral operator with singular kernel to an ${\ell }^p$ bound for all $1<p<\infty$. This requires an intricate induction argument involving layers of decompositions of the operator according to the Diophantine properties of the coefficients of its polynomial phase function.